Placini, Giovanni (2020): On the geometry and topology of Sasakian manifolds. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

PDF
Placini_Giovanni.pdf 905kB 
Abstract
This thesis is concerned with the topology and geometry of Sasakian manifolds. Sasaki structures consist of certain contact forms equipped with special Riemannian metrics. Sasakian manifolds relate to arbitrary contact manifolds as Kählerian or projective complex manifolds relate to arbitrary symplectic manifolds. Therefore, Sasakian manifolds are the odddimensional analogs of Kähler manifolds. In the first part of the thesis we discuss some geometric invariants of Sasaki structures. Specifically, the socalled basic Hodge numbers, the type and their relation to the underlying contact and almost contact structures are discussed. We produce many pairs of negative Sasakian structures with distinct basic Hodge numbers on the same differentiable manifold in any odd dimension larger than 3. In the second part of the thesis we discuss topological properties of Sasakian manifolds, focussing particularly on the fundamental groups of compact Sasakian manifolds. In parallel with the theory of Kähler and projective groups, we call these groups Sasaki groups. We prove that any projective group is realizable as the fundamental group of a compact Sasakian manifold in every odd dimension larger than three. Similarly, every finitely presentable group is realizable as the fundamental group of a compact Kcontact manifold in every odd dimension larger than three. Nevertheless, Sasaki groups satisfy some very strong constraints, some of which are reminiscent of known constraints on Kähler groups. We show that the class of Sasaki groups is not closed under direct products and that there exist Sasaki groups that cannot be realized in arbitrarily large dimension. We prove that Sasaki groups behave similarly to Kähler groups regarding their relation to 3manifold groups and to free products.
Item Type:  Theses (Dissertation, LMU Munich) 

Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 510 Mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date of oral examination:  4. August 2020 
1. Referee:  Kotschick, Dieter 
MD5 Checksum of the PDFfile:  f8f111246548d5036f66b6f6e4420590 
Signature of the printed copy:  0001/UMC 27289 
ID Code:  26523 
Deposited On:  21. Aug 2020 09:51 
Last Modified:  23. Oct 2020 13:46 